TRIANGULAR MATRIX REPRESENTATIONS OF SKEW MONOID RINGS

Let R be a ring and S a u.p.-monoid. Assume that there is a monoid homomorphism &#945; : S &#8594; Aut (R). Suppose that &#945; is weakly&#12288;rigid and lR(Ra) is pure as a left ideal of R for every element a &#8712; R. Then the skew monoid ring R*S induced by &#945; has the same triangulating dimension as R. Furthermore, if R is a PWP ring, then so is R*S.</p

FP-GR-INJECTIVE MODULES

In this paper, we give some characterizations of FP-grinjective R-modules and graded right R-modules of FP-gr-injective dimension at most n. We study the existence of FP-gr-injective envelopes and FP-gr-injective covers. We also prove that (1) (⊥gr-FI, gr-FI) is a hereditary cotorsion theory if and only if R is a left gr-coherent ring, (2) If R is right gr-coherent with FP-gr-id(RR) ≤ n, then (gr-FIn, gr-F n⊥) is a perfect cotorsion theory, (3) (⊥gr-FIn, gr-FIn) is a cotorsion theory, where gr-FI denotes the class of all FP-gr-injective left R-modules, gr-FIn is the class of all graded right R-modules of FP-gr-injective dimension at most n. Some applications are given

Radial Spokes-A Snapshot of the Motility Regulation, Assembly, and Evolution of Cilia and Flagella

Propulsive forces generated by cilia and flagella are used in events that are critical for the thriving of diverse eukaryotic organisms in their environments. Despite distinctive strokes and regulations, the majority of them adopt the 9+2 axoneme that is believed to exist in the last eukaryotic common ancestor. Only a few outliers have opted for a simpler format that forsakes the signature radial spokes and the central pair apparatus, although both are unnecessary for force generation or rhythmicity. Extensive evidence has shown that they operate as an integral system for motility control. Recent studies have made remarkable progress on the radial spoke. This review will trace how the new structural, compositional, and evolutional insights pose significant implications on flagella biology and, conversely, ciliopathy

Imaging Ferroelectric Domains via Charge Gradient Microscopy Enhanced by Principal Component Analysis

Local domain structures of ferroelectrics have been studied extensively using various modes of scanning probes at the nanoscale, including piezoresponse force microscopy (PFM) and Kelvin probe force microscopy (KPFM), though none of these techniques measure the polarization directly, and the fast formation kinetics of domains and screening charges cannot be captured by these quasi-static measurements. In this study, we used charge gradient microscopy (CGM) to image ferroelectric domains of lithium niobate based on current measured during fast scanning, and applied principal component analysis (PCA) to enhance the signal-to-noise ratio of noisy raw data. We found that the CGM signal increases linearly with the scan speed while decreases with the temperature under power-law, consistent with proposed imaging mechanisms of scraping and refilling of surface charges within domains, and polarization change across domain wall. We then, based on CGM mappings, estimated the spontaneous polarization and the density of surface charges with order of magnitude agreement with literature data. The study demonstrates that PCA is a powerful method in imaging analysis of scanning probe microscopy (SPM), with which quantitative analysis of noisy raw data becomes possible